Problem: A group of adults and kids went to see a movie. Tickets cost $$7.50$ each for adults and $$4.00$ each for kids, and the group paid $$51.00$ in total. There were $7$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Explanation: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${7.5x+4y = 51}$ ${x = y-7}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-7}$ for $x$ in the first equation. ${7.5}{(y-7)}{+ 4y = 51}$ Simplify and solve for $y$ $ 7.5y-52.5 + 4y = 51 $ $ 11.5y-52.5 = 51 $ $ 11.5y = 103.5 $ $ y = \dfrac{103.5}{11.5} $ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into ${x = y-7}$ to find $x$ ${x = }{(9)}{ - 7}$ ${x = 2}$ You can also plug ${y = 9}$ into ${7.5x+4y = 51}$ and get the same answer for $x$ ${7.5x + 4}{(9)}{= 51}$ ${x = 2}$ There were $2$ adults and $9$ kids.